Use the graph of f ( x ) f\left(x\right) f ( x ) to determine if the function is continuous or discontinuous at x = c x=c x = c . c = 3 c=3 c = 3

Here we want to use the graph of f of x to determine if our function is continuous or discontinuous at x equals c. Now here we're looking at where x is equal to 3 because that's our specified value of c. Now looking at my graph here, if I were to go ahead and trace my function where x is equal to 3, I see that I would have to pick up my pen and move it to a different place as x is c. So I know that I actually have an asymptote here, meaning that this function is a discontinuous. Now because we just did this by tracing our graph and verifying that we had to lift our pen, you may also be asked to actually verify this using values which you could easily do by finding your limit and your function value. Now here our limit as x approaches 3 of this function actually does not exist because we have unbounded values and our function value also doesn't exist. So here our function is discontinuous due to this asymptote here. Thanks for watching and let me know if you have questions.

1. Limits and Continuity

Continuity

Business Calculus

1. Limits and Continuity

Continuity

Struggling with Business Calculus?

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Multiple Choice

Use the graph of $f\left(x\right)$ to determine if the function is continuous or discontinuous at $x=c$ .
$c=3$

Continuous

Discontinuous

Verified step by step guidance

Step 1: Recall the definition of continuity. A function is continuous at a point x = c if the following three conditions are satisfied: (1) f(c) is defined, (2) the limit of f(x) as x approaches c exists, and (3) the limit of f(x) as x approaches c is equal to f(c).

Step 2: Analyze the graph at x = 3. Observe that there is a break in the graph at x = 3, indicating a potential discontinuity.

Step 3: Check if f(3) is defined. From the graph, there is no point at x = 3 on the curve, which means f(3) is not defined.

Step 4: Evaluate the limit of f(x) as x approaches 3. The graph shows that the left-hand limit and right-hand limit do not converge to the same value, indicating the limit does not exist.

Step 5: Conclude that the function is discontinuous at x = 3 because it fails the conditions for continuity: f(3) is not defined, and the limit of f(x) as x approaches 3 does not exist.